Endpoint Sobolev regularity of bilinear maximal commutators

Feng Liu; Xueying Zhu; Feng Liu; Xueying Zhu

doi:10.3934/math.2025902

AIMS Mathematics

2025, Volume 10, Issue 9: 20199-20218. doi: 10.3934/math.2025902

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Endpoint Sobolev regularity of bilinear maximal commutators

Feng Liu ^,,
Xueying Zhu

College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, Shandong Province 266590, China

Received: 05 July 2025 Revised: 26 August 2025 Accepted: 01 September 2025 Published: 04 September 2025
MSC : 42B25, 46E35

In this paper our objective of investigation was the endpoint Sobolev regularity of the bilinear maximal commutator
$ \mathfrak{M}_{b, \alpha}(f, g)(x) = \sup\limits_{r>0}\frac{1}{(2r)^{1-\alpha}}\int_{-r}^{r}|(b(x)-b(x+y))f(x+y)g(x-y)|dy, $
where $ \alpha\in[0, 1) $ and $ b\in{Lip}(\mathbb{R}) $ with $ b'\in L^1(\mathbb{R}) $. We showed that the map $ \mathfrak{M}_{b, \alpha}:W^{1, 1}(\mathbb{R})\times W^{1, 1}(\mathbb{R})\rightarrow W^{1, q}(\mathbb{R}) $ was bounded and continuous for $ q\in(\frac{1}{1-\alpha}, \infty) $. The main result essentially answered a question motivated by Wang and Liu in 2022.
- bilinear maximal commutator,
- endpoint Sobolev space,
- boundedness,
- continuity
Citation: Feng Liu, Xueying Zhu. Endpoint Sobolev regularity of bilinear maximal commutators[J]. AIMS Mathematics, 2025, 10(9): 20199-20218. doi: 10.3934/math.2025902

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Abstract

In this paper our objective of investigation was the endpoint Sobolev regularity of the bilinear maximal commutator

$ \mathfrak{M}_{b, \alpha}(f, g)(x) = \sup\limits_{r>0}\frac{1}{(2r)^{1-\alpha}}\int_{-r}^{r}|(b(x)-b(x+y))f(x+y)g(x-y)|dy, $

where $ \alpha\in[0, 1) $ and $ b\in{Lip}(\mathbb{R}) $ with $ b'\in L^1(\mathbb{R}) $. We showed that the map $ \mathfrak{M}_{b, \alpha}:W^{1, 1}(\mathbb{R})\times W^{1, 1}(\mathbb{R})\rightarrow W^{1, q}(\mathbb{R}) $ was bounded and continuous for $ q\in(\frac{1}{1-\alpha}, \infty) $. The main result essentially answered a question motivated by Wang and Liu in 2022.

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